Testing the Role of Diagonal Interactions in High-Order Hopfield Models via Dynamical Mean-Field Theory

TL;DR

This paper investigates the slow retrieval dynamics in high-order Hopfield models, demonstrating that these effects are due to intrinsic high-order interactions rather than diagonal self-interactions.

Contribution

The study shows that slow dynamics and large basins of attraction in high-order Hopfield models are caused by intrinsic high-order interactions, not diagonal self-interactions.

Findings

Slow dynamics persist even without diagonal self-interactions.

Large basins of attraction are intrinsic to high-order interactions.

Slowdowns are due to properties of high-order interactions, not diagonal contributions.

Abstract

High-order extensions of the Hopfield model are known to exhibit dramatically enhanced storage capacity at equilibrium, while their dynamical retrieval properties remain less well understood. In our previous work, we carried out a dynamical mean-field theory (DMFT) analysis of the Krotov--Hopfield-type dense associative memory and found that the transition between successful and failed retrieval is accompanied by pronounced slow dynamics. As a consequence, the effective basin of attraction observed in numerical simulations extends well beyond that predicted by equilibrium statistical mechanics. A natural hypothesis is that this discrepancy originates from diagonal (self-interaction) contributions in the Krotov--Hopfield model, which generate a large number of lower-order interaction terms and may induce glassy relaxation near the retrieval boundary. To test this hypothesis, we analyze an alternative high-order associative memory model, namely the Abbott--Arian-type $p$-body Hopfield model, in which such diagonal contributions are absent by construction. Using dynamical mean-field theory, we derive an effective single-site process together with closed macroscopic equations governing the retrieval dynamics. Our analysis reveals that both slow dynamics and a substantial enlargement of the apparent basin of attraction persist even in this model. These results indicate that the dynamical slowdown near the retrieval boundary cannot be attributed primarily to diagonal self-interaction effects, but instead originates from intrinsic properties of high-order interactions.